Thermal equilibrium of a two-dimensional model of mobile point-like charges with pair logarithmic interactions, immersed in a fixed neutralising background, is exactly solvable at a special coupling constant. Assuming gauge invariance of the statistical mechanics and the requirement of local charge neutrality in the bulk regime of the system, the exact solution implies explicit expressions for inverse elements of an infinite interaction matrix. This result is equivalent to new summation formulas over upper index for bilinear products of Jacobi and Gegenbauer polynomials which are complementary to standard completeness relations involving lower indices for these polynomials. We go beyond the physical input and derive multi-variable generalizations of the summation formulas.

具有对数相互作用的移动点状电荷二维模型的热平衡浸入固定的中和背景中，可以在特殊的耦合常数下精确求解。假设统计力学的规范不变性和系统主体区域中局部电荷中性的要求，精确解意味着无限相互作用矩阵的逆元素的显式表达式。该结果等效于Jacobi 和 Gegenbauer 多项式的双线性乘积的上指标上的新求和公式，这些公式是对涉及这些多项式下指标的标准完整性关系的补充。我们超越了物理输入并推导出求和公式的多变量概括。